Question-221268

Question Number 221268 by behi834171 last updated on 29/May/25

Commented by behi834171 last updated on 29/May/25

CF=(√2) , CD=2 , CE=3 possible value(s) for:R=? R=radius. A=center

$$\boldsymbol{{CF}}=\sqrt{\mathrm{2}}\:\:,\:\:\boldsymbol{{CD}}=\mathrm{2}\:\:\:,\:\:\:\:\boldsymbol{{CE}}=\mathrm{3} \\ $$$$\boldsymbol{{possible}}\:\boldsymbol{{value}}\left(\boldsymbol{{s}}\right)\:\boldsymbol{{for}}:\boldsymbol{{R}}=? \\ $$$$\boldsymbol{{R}}=\boldsymbol{{radius}}.\:\:\:\boldsymbol{{A}}=\boldsymbol{{center}} \\ $$

Answered by mr W last updated on 29/May/25

Commented by mr W last updated on 29/May/25

let′s construct three concentric circles with center at C and radii (√2), 2 and 3 respectively. the points F, D and E must lie on these three circles respectively. each circle, which intersects with all these three circles, like the green one, fulfills the condictions for the circle that we are searching for. we can see there are infinite such circles which intersect the three concenteric circles with radii (√2), 2 and 3. we can find the smallest one among them, such as the red one, which has the radius ((3−(√2))/2). but there doesn′t exist the largest one. therefore the possible values for R are in the range [((3−(√2))/2), ∞).

$${let}'{s}\:{construct}\:{three}\:{concentric}\: \\ $$$${circles}\:{with}\:{center}\:{at}\:\boldsymbol{{C}}\:{and}\:{radii} \\ $$$$\sqrt{\mathrm{2}},\:\mathrm{2}\:{and}\:\mathrm{3}\:{respectively}. \\ $$$${the}\:{points}\:\boldsymbol{{F}},\:\boldsymbol{{D}}\:{and}\:\boldsymbol{{E}}\:{must}\:{lie}\:{on} \\ $$$${these}\:{three}\:{circles}\:{respectively}. \\ $$$${each}\:{circle},\:{which}\:{intersects}\:{with} \\ $$$${all}\:{these}\:{three}\:{circles},\:{like}\:{the}\:{green} \\ $$$${one},\:{fulfills}\:{the}\:{condictions}\:{for} \\ $$$${the}\:{circle}\:{that}\:{we}\:{are}\:{searching}\:{for}. \\ $$$${we}\:{can}\:{see}\:{there}\:{are}\:{infinite}\:{such} \\ $$$${circles}\:{which}\:{intersect}\:{the}\:{three} \\ $$$${concenteric}\:{circles}\:{with}\:{radii}\:\sqrt{\mathrm{2}},\:\mathrm{2} \\ $$$${and}\:\mathrm{3}.\:{we}\:{can}\:{find}\:{the}\:{smallest} \\ $$$${one}\:{among}\:{them},\:{such}\:{as}\:{the}\:{red} \\ $$$${one},\:{which}\:{has}\:{the}\:{radius}\:\frac{\mathrm{3}−\sqrt{\mathrm{2}}}{\mathrm{2}}. \\ $$$${but}\:{there}\:{doesn}'{t}\:{exist}\:{the}\:{largest} \\ $$$${one}.\:{therefore}\:{the}\:{possible}\:{values} \\ $$$${for}\:{R}\:{are}\:{in}\:{the}\:{range}\:\left[\frac{\mathrm{3}−\sqrt{\mathrm{2}}}{\mathrm{2}},\:\infty\right). \\ $$

Commented by behi834171 last updated on 29/May/25

thank you very much dear master for your time. the same answer, if : C lies on the circle , or not?

$${thank}\:{you}\:{very}\:{much}\:{dear}\:{master}\:{for} \\ $$$${your}\:{time}. \\ $$$${the}\:{same}\:{answer},\:{if}\::\:\boldsymbol{{C}}\:\:{lies}\:{on}\:{the}\:{circle}\:, \\ $$$${or}\:{not}? \\ $$

Commented by mr W last updated on 29/May/25

if C should lie on the circle, we have a different minimum circle with R_(min) =(3/2), but there still doesn′t exist the maximum circle.

$${if}\:{C}\:{should}\:{lie}\:{on}\:{the}\:{circle},\:{we} \\ $$$${have}\:{a}\:{different}\:{minimum}\:{circle} \\ $$$${with}\:{R}_{{min}} =\frac{\mathrm{3}}{\mathrm{2}},\:{but}\:{there}\:{still}\:{doesn}'{t} \\ $$$${exist}\:{the}\:{maximum}\:{circle}. \\ $$

Commented by mr W last updated on 29/May/25

Facebook Tweet Pin

Leave a Reply Cancel reply